Hierarchically hyperbolic spaces II: Combination theorems and the distance formula

TL;DR

This paper develops tools for studying hierarchically hyperbolic spaces (HHS), proving a distance formula, exploring examples like 3-manifold groups, and introducing hierarchical quasiconvexity, thus broadening understanding of complex geometric structures.

Contribution

It introduces a streamlined set of axioms for HHS, proves a general distance formula, and establishes a combination theorem for trees of HHSs, expanding the class of known HHS examples.

Findings

All HHSs satisfy a Masur-Minsky-style distance formula.

A characterization of when 3-manifold groups are HHSs.

A new combination theorem for trees of HHSs.

Abstract

We introduce a number of tools for finding and studying \emph{hierarchically hyperbolic spaces (HHS)}, a rich class of spaces including mapping class groups of surfaces, Teichm\"{u}ller space with either the Teichm\"{u}ller or Weil-Petersson metrics, right-angled Artin groups, and the universal cover of any compact special cube complex. We begin by introducing a streamlined set of axioms defining an HHS. We prove that all HHSs satisfy a Masur-Minsky-style distance formula, thereby obtaining a new proof of the distance formula in the mapping class group without relying on the Masur-Minsky hierarchy machinery. We then study examples of HHSs; for instance, we prove that when $M$ is a closed irreducible $3$--manifold then $\pi_1M$ is an HHS if and only if it is neither $Nil$ nor $Sol$. We establish this by proving a general combination theorem for trees of HHSs (and graphs of HH groups). We also introduce a notion of "hierarchical quasiconvexity", which in the study of HHS is analogous to the role played by quasiconvexity in the study of Gromov-hyperbolic spaces.